This seems to be slicing the onion just a biiiiit too thin. Between War domain Clerics and Paladins...you want to sandwich a full caster (cleric) with more...[martial-ity? Sure, why not?] ...martiality than a cleric, but not so much as a paladin....all the while allowing them to accrue the powers of TWO separate subclasses, their Domain stuff AND an Oath. And, I am guessing from the description, they are allowed access to both Cleric and Paladin spell lists (to give them access to spell-smites).
It seems...completely unnecessary, as a concept. In implementation, it's WILDLY overpowered.
But that's not why we're here.
My two thoughts for developing an "Inquisitor" classed character are:
1. "Inquisitor" may be better/best served simply by making it a clerical Domain, instead of its own "full" class. Making it its own class then begs the question, "What are its subclasses?" Apparently it is supposed to pick a Divine Domain AND, then, a Sacred Oath. They're just picking from the PHB lists here? Create a Divine Domain of..."Inquisition"...or, maybe, define/flavor it as "Truth" or somesuch. Throw in a couple of smite spells in their Domain spell list. Give them a paladin-esque -but unique!- "aura" power as their Domain Channel ability, and a couple of -if its what you're looking for- badass martial stunts/abilities as the domain features.
2. My other thought would be, if a "hybrid/all its own full class" is what you want...an Inquisitor strikes me as more of a Cleric/Rogue mix...or, maybe even, Cleric/Ranger. Give them tracking abilities (and/or magic) to hunt down the enemies of their religion. Some stealth and perception boosts, for sure. A magical/divine smite ability probably wouldn't be out of line...makes them the only folks other than paladins that can do that.
2a. By virtue of creating them as a "hybrid" class, it kind of demands they be taken to "half caster." They are a Cleric that also has some Rogue/Assassin/Ranger tricks. Keeping their Full caster status is unnecessarily unbalanced/overpowered. You're "splitting", in effect, the classes that compose the hybrid. The class gets pieces of each, not "full" of either.
2b. As mentioned above, if you go the "its own full class" route, in 5e, then you need to define subclass flavors/archetypes that fill that class' cup. Per the PHB, it would seem you need a minimum of 2, preferably 3, to justify the Inquisitor's Existence as its own class. Off the top of my head, I would begin with something like "Justiciar" for the more martially-inclined wandering judge/executioner style guy, and, let's call it, "Witch Hunter" for a more magically-inclined [judge/executioner] supernaturally adept guy. I'm hard pressed to come up with a third character archetype that fits the name and flavor of an "Inquisitor", other than a more specifically ranger-esque "track 'em down wherever they go/can't escape me" kind of fellow.
So, yeah. At the very least, with what you've got, if they're getting 2 subclasses, they shouldn't be "full casters" and vice versa. And for my two cents, I think a Domain is best way to go.
